The problem asks to match the compositions of two functions $f(x) = 3x + 5$ and $g(x) = 2x^2 + 3x$ with their equivalent expressions.

We are given four options for compositions of functions:

1. $g \circ f$
2. $f \circ f$
3. $g \circ g$
4. $f \circ g$

Let's calculate each composition:

1. $g \circ f(x) = g(f(x))$
   We first substitute $f(x) = 3x + 5$ into $g(x) = 2x^2 + 3x$.
   $g(f(x)) = 2(3x + 5)^2 + 3(3x + 5)$ First expand $(3x + 5)^2 = 9x^2 + 30x + 25$:
   $g(f(x)) = 2(9x^2 + 30x + 25) + 3(3x + 5)$ Expand both terms: $= 18x^2 + 60x + 50 + 9x + 15$ Simplifying: $g(f(x)) = 18x^2 + 69x + 65$ This corresponds to option D.

2. $f \circ f(x) = f(f(x))$
   Substitute $f(x) = 3x + 5$ into itself: $f(f(x)) = 3(3x + 5) + 5$ Expand: $= 9x + 15 + 5 = 9x + 20$ This corresponds to option B.

3. $g \circ g(x) = g(g(x))$
   Substitute $g(x) = 2x^2 + 3x$ into itself: $g(g(x)) = 2(2x^2 + 3x)^2 + 3(2x^2 + 3x)$ First, expand $(2x^2 + 3x)^2 = 4x^4 + 12x^3 + 9x^2$:
   $g(g(x)) = 2(4x^4 + 12x^3 + 9x^2) + 3(2x^2 + 3x)$ Expand both terms: $= 8x^4 + 24x^3 + 18x^2 + 6x^2 + 9x$ Simplifying: $g(g(x)) = 8x^4 + 24x^3 + 24x^2 + 9x$ This corresponds to option A.

4. $f \circ g(x) = f(g(x))$
   Substitute $g(x) = 2x^2 + 3x$ into $f(x) = 3x + 5$: $f(g(x)) = 3(2x^2 + 3x) + 5$ Expand: $= 6x^2 + 9x + 5$ This corresponds to option C.

Final Answers:

1. $g \circ f$ → D
2. $f \circ f$ → B
3. $g \circ g$ → A
4. $f \circ g$ → C

Let me know if you need further explanation or have any questions!

Here are 5 related questions for practice:

1. What is the inverse of $f(x) = 3x + 5$?
2. Can you find the inverse of $g(x) = 2x^2 + 3x$?
3. What happens if you compose two inverse functions?
4. How do you verify if two functions are inverses?
5. What is the domain and range of the function $g(x) = 2x^2 + 3x$?

Tip: When composing functions, always remember to substitute one function into the other and carefully simplify the expression step by step!